Section 12.1 - Double Integrals over rectangles
 We want to find the volume of the space bounded by the graph of fx, y ≥ 0 and the xy-plane over
the rectangular region
R  a, b  c, d  x, y | a ≤ x ≤ b, c ≤ y ≤ d
We first subdivide R into subrectangles by dividing the interval a, b into m subintervals x i−1 , x i  of
equal width Δx  b−a
m and dividing the interval c, d into n subintervals y j−1 , y j  of equal width
d−c
Δy  n . This creates mn subrectangles of area ΔA  ΔxΔy. For an arbitrary subrectangle
R ij  x i−1 , x i   y j−1 , y j   x, y | x i−1 ≤ x ≤ x i , y j−1 ≤ y ≤ y j 
we choose a sample point x ∗ij , y ∗ij . Then we can approximate the volume of space below f over the
region R ij by the rectangular box (or "column") with base R ij and height fx ∗ij , y ∗ij . The volume of
such a box is
ΔV ij  fx ∗ij , y ∗ij ΔA
If we follow this procedure for all the subrectangles and add the corresponding volumes, we get an
estimate for the desired volume
m
V≈
n
m
n
∑ ∑ ΔV ij  ∑ ∑ fx ∗ij , y ∗ij ΔA
i1 j1
i1 j1
If we take the limit as m, n →  we get the desired volume
m
V  m,n→
lim
n
∑ ∑ fx ∗ij , y ∗ij  ΔA
i1 j1
 The double integral of fx, y over the rectangular area R is
m
n
lim ∑ ∑ fx ∗ij , y ∗ij  ΔA
 R fx, y dA  m,n→
i1 j1
 It can be shown that the limits above exist if f is continuos on R (they also exists for some
functions that are not continuous on R).
 If fx, y ≥ 0 on R, then the double integral represents the volume of the space bounded by the
graph of f and the xy-plane over R.
 For estimating we can choose the sample points x ∗ij , y ∗ij  to be any point in the subrectangle R ij .
Common choices are the upper right corner, lower right corner, upper left corner, lower left
corner, or midpoint of rectangle R ij . If we choose the upper right corner then we have
x ∗ij , y ∗ij   x i , y j .
 Examples:
1. Estimate the volume of the surface that lies below the graph of fx, y  x 2  y 2  2 over the
rectangular region x, y | 0 ≤ x ≤ 4, 0 ≤ y ≤ 2 with m  4 and n  2. Do it using the upper
right corner of each subrectangle for the sample point.
1
2. Estimate the value of  fx, y dA if the contour plot of f is as shown below and
R
R  −3, 3  −4, 4. Use Midpoints of the subrectangles with m  n  2.
5
4
3
2
1
-5
-4
-3
-2
1
-1
-1
1
2
2
3
3
-2
4
5
4
5
-3
-4
6
-5
3. Find the exact value of  |x − 2| dA if R  x, y | 0 ≤ x ≤ 2, 0 ≤ y ≤ 3
R
Homework: Page 847 #’s 3, 9 (for part (b) see page 844 for a discussion of average value),
11, 12, 13
 More practice. Turn the below problems in for a grade. Due by Thursday March 23. Show all work
on separate paper. Worth 6 points.
1. Estimate the value of  fx, y dA in each case:
R
a. fx, y  x  2y  3, R  x, y | 0 ≤ x ≤ 3, −1 ≤ y ≤ 1, with m  3 & n  4 using the
upper right corners of subrectangles for sample points.
b. fx, y  x  2y  3, R  x, y | 0 ≤ x ≤ 3, −1 ≤ y ≤ 1, with m  3 & n  4 using the
lower left corners of subrectangles for sample points.
c. fx, y  x  2y  3, R  x, y | 0 ≤ x ≤ 3, −1 ≤ y ≤ 1, with m  3 & n  2 using the
midpoints of subrectangles for sample points.
2. Find the exact value of the double integral by identifying it as the volume of a solid:
a.
b.
c.
 R 4 − y 2 dA if R  x, y | 0 ≤ x ≤ 2, − 2 ≤ y ≤ 2
 R 4 − y 2 dA if R  x, y | 0 ≤ x ≤ 2, 0 ≤ y ≤ 2
 R −|x|  2 dA if R  x, y | − 2 ≤ x ≤ 2, 0 ≤ y ≤ 3
2